FINITELY DEEP MATRICES

JOHN FAULKNER AND KEVIN MCCRIMMON

Dedicated to Ivan Shestakov on the occasion of his 60th birthday

Abstract. In the algebraic study of deep matrices DM_{X}(𝔽) on a finite set of indices over a field, Christopher Kennedy has recently shown that there is a unique proper ideal Z whose quotient is a central simple algebra. He showed that this ideal, which doesn’t appear for infinite index sets, is itself a central simple algebra. In this paper we extend the result to deep matrices with a finite set of 2 or more indices over an arbitrary coordinate algebra A, showing that when the coordinates are simple there is again such a unique proper ideal, and in general that the lattice of ideals of DM_{X}(A)/Z and Z are isomorphic to the lattice of ideals of the coordinate algebra A.

In [1, 1.13, p,179] J. Cuntz introduced C^{∗}-algebras O_{n} of operators on a separable Hilbert space generated by a finite family of n orthogonal isometries S_{i}, S_{i}^{∗}, subject to S_{i}^{∗}S_{j} = δ_{ij}1 and the additional condition ∑_{i=1}^{n} S_{i}S_{i}^{∗} = 1, and showed by analytic methods that these algebras are simple. Recently, C. Kennedy [4] showed that the condition ∑_{i}^{n} x_{i}x_{i}^{∗} = 1 can be removed in the finite case: there is a unique proper ideal Z in the deep matrix algebra DM_{X}(𝔽) over any field 𝔽, which is itself a simple algebra of finite matrices, and that imposing the condition ∑_{i}^{n} x_{i}x_{i}^{∗} = 1 is equivalent to passing to the quotient DM_{X}(𝔽)/Z.

1. Deep Matrix Algebras

We recall the definition of the algebra of deep matrices [6] with coefficients in a unital associative (but usually noncommutative) algebra A over some fixed ring of scalars Φ. For any nonempty index set X the set of deep X-indices or heads is the free monoid H(X) := Mon(X) based on X, consisting of all finite n-tuples h = (x_{1}, …, x_{n}) of arbitrary depth |h| = n ⩾ 0 whose individual entries x_{i} come from X; the unit element of this monoid is the empty-tuple ∅ of depth 0. The heads carry a partial order h ⊩ k if h begins k, i.e., k = hk′ begins with h, in which case we say k is deeper than h; h is a proper head of k, h ⊢ k, if k′ ≠ ∅. Two heads h, k are related h ∼ k if one is deeper than the other, h ⊩ k or k ⊩ h, otherwise they are unrelated h ≁ k. If h ∼ k, the question of who is deeper is strictly a matter of depth and, as a consequence, two heads of the same depth are related iff they are equal.

Depth: h ∼ k, |h| ⩽ |k| ⟺ h ⊩ k; h ∼ k, |h| = |k| ⟺ h = k.

Research partially supported by the Spanish Ministerio de Educació y Ciencia MTM2004-06580-C02-01 and Fondos FEDER. The authors thank the referee for helpful expository suggestions .

The set of bodies B(X) := ∏_{1}^{∞} X consists of all infinite sequences b = (y_{1}, y_{2}, …); if |X| = 1 there is only one body (x, x, x, …), but if |X| ⩾ 2 there are uncountably many bodies. We say the head h = (x_{1}, …, x_{n}) begins a body b, h ⊢ b, if (x_{1}, …, x_{n}) = (y_{1}, …, y_{n}), i.e., b = hb′ = (x_{1}, …, x_{n}, y_{n+1}, y_{n+2}, …). In this case we say that h is the nth head h_{n}(b) of b. If b_{1}, …, b_{r} are distinct bodies we have Head Separation [6, Th.20.1(iii), p.265]: there is an N so that the heads h_{N}(b_{1}), … h_{N}(b_{r}) are distinct [if b_{i}, b_{j} differ in the N_{ij} entry let N = max_{i≠j} N_{ij}].

The deep matrix algebra based on X over A is spanned by all deep matrix units E_{h,k} where the row and column indices are heads in H(X). We will call the elements of A scalars and for clarity denote them by lowercase Greek α, β, γ, whereas elements of the matrix algebra (linear combinations of matrix units) will be denoted by lowercase Latin u, v, w. The notation E_{h,k} [4] (E_{h}^{k} in [6]) is chosen to emphasize the analogy with ordinary matrix units E_{i,j}.

Rather than this matrix notation, we will usually use the C^{∗}-algebra notation denoting the basis elements by “back-and-forth shifts” hk^{∗} (segregated monoid products with h ∈ H(X) at the beginning and k^{∗} ∈ H(X^{∗}) at the end), which are elements of the free monoid-with-involution H(X, ∗) := Mon(X ⊔ X^{∗}) for set X and a disjoint copy X^{∗}. The map ∗ induces a reversal involution on this monoid via (x^{∗})^{∗} = x, and the unit of the monoid (the empty product of x’s and also the empty product of x^{∗}’s) is denoted simply 1 instead of ∅. We refer to the elements of H(X) as “forward shifts” and those of H(X^{∗}) as “backward shifts” of “negative depth”, leading us to define a general notion of depth for back-and-forth shifts in H(X, ∗) (distinguished by ∥ in place of |):

∥u∥ = ∥hk^{∗}∥ := |h| − |k|, ∥u^{∗}∥ = −∥u∥, ∥k^{∗}∥ := −|k|.

The forward and backward shifts h, k^{∗} for h, k ∈ H(X) are generated by primitive shifts x, y^{∗} for x, y ∈ X: h = (x_{1} … x_{n}) = x_{1} ⋯ x_{n}, k^{∗} = (y_{1} … y_{m})^{∗} = y_{m}^{∗} ⋯ y_{1}^{∗}. The shift monoid ShM(X) on X is the ∗-monoid-with-zero Mon(X ⊔ X^{∗} ⊔ 0) on X modulo the homogeneous relations

x^{∗}x = 1,  x^{∗}y = 0,  x0 = x^{∗}0 = 0 = 0^{∗} = 0x^{∗} = 0x (y ≠ x ∈ X).

Definition 1.1 (Deep Matrix Construction [6, Thm.2.1], [4]). The deep matrix algebra DM_{X}(A)(called E(X, A) in [6]) based on X over a unital associative coordinate algebra A consists of the free left A-module with the basis of all deep matrix units E_{h,k} for h, i, j, k ∈ H(X), determined by the Deep Multiplication Rules [1, p.175] for the basic products (α, β ∈ A) :

(DMI) (αE_{h,i})(βE_{j,k}) = (αE_{h,i})(βE_{ij′,k}) = αβE_{hj′,k} if i ⊩ j = ij′,
(DMII) (αE_{h,i})(βE_{j,k}) = (αE_{h,ji′})(βE_{j,k}) = αβE_{h,ki′} if j ⊩ i = ji′,
(DMIII) (αE_{h,i})(βE_{j,k}) = 0 if i ≁ j are unrelated.

In C^{∗}-notation DM_{X}(A) is the free A-module with basis of all hk^{∗} with rules

(DMI) (αhi^{∗})(βjk^{∗}) = (αh i^{∗})(βij′k^{∗}) = αβ(hj′)k^{∗} if i ⊩ j = ij′,
(DMII) (αhi^{∗})(βjk^{∗}) = (αh(ji′)^{∗})(βjk^{∗}) = αβh(ki′)^{∗} if j ⊩ i = ji′,
(DMIII) (αhi^{∗})(βjk^{∗}) = 0 if i ≁ j are unrelated.

The Deep Multiplication Rules follow from the Orthogonality Relations y^{∗}x = δ_{y,x}1 since these allow us to repeatedly remove any y^{∗} to the left of an x in any mixed product of x’s and y^{∗}’s until we reach a segregated shift:

Head Product Rules

The un-segregated product k^{∗}h is always zero or a forward or backward shift:

k^{∗}h = 1, ∥k^{∗}h∥ = 0,  if k = h;
k^{∗}h = h′, ∥k^{∗}h∥ = |h| − |k| > 0, if k ⊢ h = kh′;
k^{∗}h = k′^{∗}, ∥k^{∗}h∥ = |h| − |k| < 0, if h ⊢ k = hk′;
k^{∗}h = 0,  if k, h are unrelated.

The deep matrices form a unital associative algebra with unit 1 (or 1_{deep} = E_{∅,∅} in matrix notation): it is just the monoid algebra over A on the monoid-with-zero ShM(X). In analogy with group algebras k[G] = kG, this suggests a notation DM_{X}(A) = AHH^{∗} for the deep matrix algebra. The subalgebras AH, AH^{∗} are just the usual monoid algebras over A on the monoids H(X), H(X^{∗}).

The deep matrix construction yields a bifunctor from sets and coordinate algebras to unital associative algebras. The transpose map αE_{h,k} ⟶ αE_{k,h} is an isomorphism of DM_{X}(A) with DM_{X}(A^{op}). Thus, despite the apparent asymmetry in the indices, there is a duality: all general results for deep matrices over all A remain true if the roles of h, k are interchanged. If the coordinate algebra A carries an involution α → ᾱ (e.g. if A is commutative, ᾱ = α), then DM_{X}(A) carries a natural conjugate transpose involution uniquely determined by (αh k^{∗})^{∗} := ᾱk h^{∗}, yielding a functor from sets and coordinate ∗-algebras to unital ∗-algebras.

The associativity of deep matrix multiplication also follows from the faithful Kennedy representation [4] of D := DM_{X}(A) on the Kennedy module, the free right A-module

V [H] = HA := ⨁_{j∈H(X)} jA, (αh i^{∗})(jβ) = \begin{cases}hj′αβ & if i ⊩ j = ij′, \\ 0 & if i ⊮ j. \end{cases}

This is not a left regular representation and HA ≅ AH is not a D-submodule of D = AHH^{∗} since x^{∗}(1) = 0 ≠ x^{∗}.

Any deep matrix algebra DM_{X}(A) has a Frankenstein representation [6, 20.5, p.269] on the Frankenstein module, the free right A-module

V [B] = BA := ⨁_{b∈B(X)} bA

(called V (X, A) in [6]) with basis of all bodies, given by

if |i| = n then (α hi^{∗})(b β) = \begin{cases}hb′αβ & if i ⊩ b = ib′, i.e., i = h_{n}(b); \\ 0 & if i ⊮ b, i.e., i ≠ h_{n}(b). \end{cases}

The corresponding operators F_{h,i} of left multiplication by E_{h,i} = hi^{∗} chop the head i off b and replace it with h (killing the patient b if it doesn’t begin with the head i). Linear combinations of these operators form the Frankenstein algebra F_{X}(A), a left A-module (free if X is infinite) with a canonical Deep Matrix epimorphism

DM_{X}(A) ⟶^{F} F_{X}(A) via E_{h,k} → F_{h,k}.

However, we will see that F is an isomorphism (the Frankenstein representation is faithful) only when X is infinite.

The always-faithful Kennedy representation on V [H] is more satisfying than the faithless Frankenstein representation on V [B] for finite X. But the Frankenstein action actually becomes faithful if we move to a slightly larger set X′ := X ⊔ {z} (without going all the way to an infinite set): both the Frankenstein representation on B(X′) and the regular representation on DM_{X′}(A) contain many copies of the Kennedy representation V [H] (and hence are faithful).

Theorem 1.2 (Kennedy Imbedded). For arbitrary X, set X′ := X ⊔ {z}, b′ ∈ B′ := B(X′), D′ := DM_{X′}(A), D := DM_{X}(A). The Kennedy representation of D on V [H] is isomorphic to a subrepresentation of the left regular representation of D′ on W = AHz ⊆ D′, and to a subrepresentation of the Frankenstein representation of D′ on W = Hzb′A ⊆ V [B′] for each b′ ∈ B′.

Proof. We imbed V [H] ↪ AHz ⊆ D′ via the map hα = (x_{1}, x_{2}, …, x_{n})α ↦ αhz = α(x_{1}, x_{2}, …, x_{n}, z) and V [H] ↪ Hzb′ ⊆ V [B′] via the mapping hα ↦ hzb′α = (x_{1}, x_{2}, …, x_{n}, z, x′_{1}, x′_{2}, …)α. To see these W are D-invariant with Kennedy action, note first that αhi^{∗} kills βjz and jzb′β if i ⊮ jz (i.e., i ⊮ j) [we never have jz ⊩ i as in (DMII) because of the factor z ∉ X]. Next, if i ⊩ j = ij′ then as in (DMI) we have (αhi^{∗})(βjz) = (αhi^{∗})(βij′z) = αβhj′z and (αhi^{∗})(jzb′β) = (αhi^{∗})(ij′zb′β) = hj′zb′αβ.

The standard argument [4, Th.2 p.527], [6, (3) p.271]^{1} that D acts faithfully on V [B] when X is infinite merely requires a z distinct from the x’s, so it shows D acts faithfully on AHz and Hzb′A. □

The deep matrix units commute with scalar multiplication (though A itself need not be commutative)

α(hk^{∗}) = (α1)hk^{∗} = hk^{∗}(α1).

This shows that we can represent deep matrices over A as a “scalar extension” of the deep matrices over the scalar ground ring Φ,

1If (∑ α_{h,k}hk^{∗})Hz = 0 for distinct (h, k) and nonzero α_{h,k}, choose |k_{0}| minimal and then |h_{0}| maximal among all |h| among the (h, k_{0}). Then the Head Product Rules yields the contradiction 0 = h_{0}^{∗}(∑ α_{h,k}hk^{∗})k_{0}z = α_{h_{0},k_{0}}1 [never k ⊢ k_{0}, and never h_{0} ⊢ h among the (h, k_{0}) and x^{∗}z = 0].

DM_{X}(A) ≅ A ⨂_{Φ} DM_{X}(Φ).

For coordinate algebras A over a field Φ = 𝔽, this shows that several of our results can be obtained directly from Kennedy’s results for A = 𝔽.

Proposition 1.3. The deep matrices D := DM_{X}(A) have a ℤ-grading

D = ⨁_{n=−∞}^{∞} D_{n}, D_{n}D_{m} ⊆ D_{n+m}, for D_{n} := ∑_{∥hk^{∗}∥=n} A hk^{∗}.

If H_{n} denotes the heads of depth n and for N = |X_{n}|, M = |H_{m}|, then in terms of the subalgebra D_{0} we can write D_{n} := H_{n}D_{0}, D_{−n} = D_{0}H_{n}^{∗} if n > 0. The subalgebra D_{0} is a module direct sum of level-n matrix subalgebras [1, 1.4 p.175],

D_{0} = ⨁_{n=0}^{∞} D_{0}^{(n)}, D_{0}^{(n)} := ∑_{|h|=|k|=n} AE_{h,k} ≅ Mat_{N×N}^{fin}(A), D_{0}^{(n)}D_{0}^{(m)} ⊆ D_{0}^{(max{n,m})}.

If |X| > 1 the Frankenstein representation of the level-n matrix subalgebra D_{0}^{(n)} is the direct sum of uncountably many copies of the standard representation of Mat_{N×N}^{fin}(A) on ⨁^{N} A :

V [B] = H_{n}BA ≅ ⨁_{b∈B(X)} V [H_{n}]^{b}, V [H_{n}]^{b} := H_{n}bA with E_{h,i}(jb) = δ_{i,j}hb,

while the D_{n} for n ≠ 0 act as Mat_{M×N}^{fin}(A) from ⨁^{N} A to ⨁^{M} A :

if |h| = m, |k| = n, then E_{h,k}(V [H_{n}]^{b}) ⊆ V [H_{m}]^{b} with E_{h,i}(jb) = δ_{i,j}hb.

Proof. The grading decomposition comes directly from the direct decomposition into matrix units E_{h,k} = hk^{∗}, and D_{n}D_{m} ⊆ D_{n+m} follows since by the Head Product Rules if jk^{∗} hi^{∗} = jui^{∗} is nonzero then

\begin{aligned}∥jui^{∗}∥ & = |j| + |u| − |i| = |j| + (|h| − |k|) − |i| \\ & = (|j| − |k|) + (|h| − |i|) = ∥jk^{∗}∥ + ∥hi^{∗}∥. \end{aligned}

For the description of D_{n} in terms of D_{0}, note that

x_{1} ⋯ x_{i} · y_{1}^{∗} ⋯ y_{j}^{∗} = \begin{cases}x_{1} ⋯ x_{n}(x_{n+1} ⋯ x_{n+j} · y_{1}^{∗} ⋯ y_{j}^{∗}) & if i = n + j ⩾ j \\ (x_{1} ⋯ x_{i} · y_{1}^{∗} ⋯ y_{i}^{∗})y_{i+1}^{∗} ⋯ y_{i+n}^{∗} & if i ⩽ j = i + n. \end{cases}

That D_{0}^{(n)} is a matrix algebra follows from the Head Products since if |h| = |i| = |j| = |k| = n we have E_{h,i}E_{j,k} = δ_{i,j}E_{h,k}, For the products, if |h| = |i| = n, |j| = |k| = m a nonzero E_{h,i}E_{j,k} has one of the forms

(hi^{∗})(jk^{∗}) = \begin{cases}(hj′)k^{∗} = E_{hj′,k} ∈ D_{0}^{(m)} for |j′| = m − n & if n ⩽ m \\ h(i′^{∗}k^{∗}) = E_{h,ki′} ∈ D_{0}^{(n)} for |i′| = n − m & if n ⩾ m \end{cases} .

The action of the E_{h,i} on V [H_{n}]^{b} follows directly from the action (αhi^{∗})(jbβ) = δ_{i,j}hbαβ if |i| = |j| = n. □

The term “deep matrices” comes from the fact that D contains matrix subalgebras D_{0}^{(n)} isomorphic to |X^{n}| × |X^{n}| matrices with finitely many nonzero entries, and as we descend we pick up larger and larger algebras of matrices at every depth.^{2}

We mention several consequences of the Deep Multiplication Rules that will be useful in what follows. The forward shifts are left-invertible, h^{∗}h = 1, but not invertible: the range projections e_{h} := hh^{∗} form a nested family of idempotents where unrelated members are orthogonal, but shallow members contain the deeper members:

(D_{H}) \begin{matrix}e_{h}h = h, h^{∗}e_{h} = h^{∗}, \\ h ≁ k ⟹ e_{h}(kj^{∗}) = (jk^{∗})e_{h} = 0, e_{h}e_{k} = e_{k}e_{h} = 0, \\ h ⊩ k = hk′ ⟹ e_{h}(kg^{∗}) = kg^{∗}, (gk^{∗})e_{h} = gk^{∗}, e_{h}e_{k} = e_{k}e_{h} = e_{k} \\ k ⊩ h = kh′ ⟹ e_{h}(kg^{∗}) = h(gh′)^{∗}, (gk^{∗})e_{h} = (gh′)h^{∗}, e_{h}e_{k} = e_{k}e_{h} = e_{h}. \end{matrix}

In particular the e_{x} = xx^{∗} for x ∈ X form an orthogonal family of “maximal” idempotents which pick out terms beginning with x or ending with x^{∗}: e_{x}[(yh′)k^{∗}] = δ_{xy}(yh′)k^{∗} and dually [k(yh′)^{∗}]e_{x} = δ_{xy}k(yh′)^{∗}, so

(D_{X}) e_{x}e_{y} = δ_{xy}e_{x}, e_{x}(hk^{∗}) = δ_{xy}hk^{∗}, (kh^{∗})e_{x} = δ_{xy}kh^{∗} if h = yh′.

Head Separation leads [6, Th.20.9, p.272] to :

Body Separation: if b_{1}, …, b_{r} are distinct bodies there exists a projection
e_{h} = hh^{∗} with e_{h}(b_{1}) = b_{1}, e_{h}(b_{i}) = 0 for i > 1.

Remark 1.4. The principal left ideal generated by the deep matrix unit hk^{∗} is the k^{th} column ideal

De_{k} = Dk^{∗} = Dhk^{∗} = ∑_{i,j} Ai(kj)^{∗} =: AHH^{∗}k^{∗},

and the principal right ideal generated by h k^{∗} is the h^{th} row ideal

e_{h}D = hD = hk^{∗}D = ∑_{j,i} A(hj)i^{∗} =: AhHH^{∗}.

Any h ∈ H determines a “downward” push monomorphism μ_{h}(u) = h u h^{∗} sending 1 to e_{h} and hence D = 1 D 1 to the Peirce subalgebra hDh^{∗} = e_{h}De_{h}, with left inverse the “upward” pull epimorphism μ_{h}^{∗} := μ_{h^{∗}}.

Proof. Indeed, the Deep Multiplication Rules show every left or right multiple ij^{∗}hk^{∗} or hk^{∗}ij^{∗} has the above form, and conversely all such matrix units arise as multiples: i(kj)^{∗} = i(hj)^{∗}hk^{∗} and (hj)i^{∗} = hk^{∗}(kj)i^{∗}. For the push and pull, any h has h^{∗}h = 1, hh^{∗} = e_{h} so μ_{h}^{∗} ∘ μ_{h} = μ_{h^{∗}h} = 1_{D}, μ_{h} ∘ μ_{h}^{∗} = μ_{hh^{∗}} = μ_{e_{h}}. □

2When ∑_{i} x_{i}x_{i}^{∗} = 1 the D_{0}^{(n)} (denoted F_{n} in [2, Prop.2.4 p.253]) have D_{0}^{(n)} ⊆ D_{0}^{(n+1)} since hk^{∗} = h1k^{∗} = ∑_{i} hx_{i}x_{i}^{∗}k_{i}^{∗}.

2. The Obstacle

So far all our remarks apply to deep matrices over any set X. From now on we examine the special properties of finite index sets. When |X| < ∞, a nonzero element u need not have a nonzero “scalar multiple” vuw = α1, and it is not true that all ideals come from coordinate ideals: there is a rogue ideal of obstinately finite character, and until the deep matrix algebra is purged of this obstacle it cannot enjoy the desirable properties of its infinite brethren.

In the ordinary matrix algebra Mat_{n}^{fin}(A) the unit is 1 = ∑_{i} E_{ii}. In the finite

index case, by (D_{X}) e := ∑_{x∈X} e_{x} = ∑_{x∈X} E_{x,x} = ∑_{x∈X} xx^{∗}

exists in DM_{X}(A) as an idempotent, the finite idempotent, but it is NOT the unit 1; its evil twin f := 1 − e is the infinite idempotent. Kennedy [4] pointed out that e should be the “true” unit (the C^{∗}-algebraists decree e = 1 as part of their axioms for the algebra O_{n}), and the imposter f gives rise to all the difficulties in the finite case.

Note that eh = h, h^{∗}e = h^{∗} for all nonempty heads h since if h begins with x ≠ y then e_{x}h = h, e_{y}h = 0 by (D_{X}). Consequently the element f has the following multiplication rules:

(F 1) eh = h, h^{∗}e = h^{∗}, f(hk^{∗}) = (kh^{∗})f = 0 if h ≠ 1,
(F 2) for all bodies b, eb = b, fb = 0,
(F 3) f 1 f = f, f(h k^{∗})f = 0 if (h, k) ≠ (1, 1).

The obstacle is the ideal Z_{X}(A) := DfD generated by f. For finite X the reduced deep matrix algebra \overline{DM_{X}(A)} is obtained by adding the axiom ∑_{x} e_{x} = 1 (i.e., f = 0) to the Deep Multiplication Rules, equivalently, by forming the quotient

\overline{DM_{X}(A)} := DM_{X}(A)/Z_{X}(A) :  ∑_{x∈X} \overline{e_{x}} = ¯1.

A simple but useful property is the

Nonzero f-Scalar Rule α ≠ 0 ⟹ αf ≠ 0

since in fact αu = 0 forces u = 0 in any free A-module as soon as u is “monic” (has one of its basic coefficients equal to 1).

We will see shortly that Z_{X}(A) is precisely the annihilator of the Frankenstein module V [B] when |X| > 1, but for now it is immediate that it is contained in the annihilator: certainly the element f annihilates V [B] by (F2), and the annihilator is always an ideal, so it contains DfD = Z_{X}(A). This gives us an important rule:

Non-Z Scalar Rule α ≠ 0 ⟹ α1 ∉ Z_{X}(A) ⊆ Ann(V [B])

since α ≠ 0 does not annihilate the free right module V [B] [note (α1)b = bα]. Theorem 2.1 (Obstacle [4]). For a finite set X the obstacle Z_{X}(A) is a free A-module spanned by all

F-Matrix Units F_{h,k} := hfk^{∗} = hk^{∗} − ∑_{x∈X} h xx^{∗} k^{∗} (h, k ∈ H)

satisfying [1, Prop. 3.1 p.183] the

F-Matrix Unit Rules  F_{h,i}F_{j,k} = δ_{i,j}F_{h,k}  (h, i, j, k ∈ H).

This algebra is isomorphic to Mat_{∞}^{fin}(A), the countably-infinite square matrices with only a finite number of nonzero entries from A, under an isomorphism sending f to the matrix unit E_{00}. The lattice of ideals of the obstacle is the lattice of ideals of A: every ideal in Z_{X}(A) is Z_{X}(I) = IZ_{X}(A) for an ideal I ◁ A. In particular, Z_{X}(A) is simple iff A is simple.

The centroid of the obstacle is the center of A: Γ(Z_{X}(A)) = C(A)1_{Z} (of course Z_{X}(A) has no center) .

Proof. For the F -matrix unit rules, if i ≠ j then i^{∗}j ≠ 1 by the Head Product Rule, so by (F3) f i^{∗}j f = 0. If i = j we have hf i^{∗}j fk^{∗} = hf 1 fk^{∗} = hfk^{∗}. Thus in all cases

F_{h,i}F_{j,k} = (hfi^{∗})(jfk^{∗}) = δ_{i,j}hfk^{∗} = δ_{i,j}F_{h,k}.

This is the multiplication table for the free left A-module Mat_{∞}^{fin}(A) with standard matrix units E_{m,n} for m, n ∈ ℕ_{0} := ℕ ∪ {0}, and if we label the countable number of heads as h_{m}, m ∈ ℕ_{0} with h_{0} = 1, there is a natural A-linear epimorphism

Mat_{∞}^{fin}(A) ⟶^{π} Z_{X}(A) via E_{m,n} ↦ F_{h_{m},h_{n}},

where E_{0,0} is sent to F_{h_{0},h_{0}} = F_{0,0} = f. This map is an isomorphism (equivalently, Z_{X}(A) is free with basis of F_{h,k}) since ∑_{h,k} α_{h,k}F_{h,k} = 0 with some α_{i,j} ≠ 0 implies 0 = F_{0,i}( ∑_{h,k} α_{h,k}F_{h,k})F_{j,0} = α_{i,j}F_{0,0} = α_{i,j}f, contrary to the Nonzero f-Scalar Rule.

From another viewpoint, the above argument shows that every ideal I of Mat_{∞}^{fin}(A) is Mat_{∞}^{fin}(I) for I = {α ∈ A | αE_{m,n} ∈ I for some (hence all) m, n}. The kernel of π must therefore be Mat_{∞}^{fin}(I) for some ideal I ◁ A, in particular it contains IE_{0,0}, so the image If = π(IE_{0,0}) must vanish. But by the Nonzero f-Scalar Rule this forces I = 0, so π is an isomorphism.

It is well-known that the center is 0 (there is no unit element in Mat_{∞}^{fin}(A) since the identity matrix has an infinite number of nonzero entries), and that the centroid is just multiplication by central elements of A. □

The Peirce decomposition of D and the obstacle Z = DfD relative to the idempotent e is

D = D_{11} ⊕ D_{10} ⊕ D_{01} ⊕ D_{00}, Z_{X}(A) = D_{10}D_{01} ⊕ D_{10} ⊕ D_{01} ⊕ D_{00}.

Here the Peirce spaces D_{10}D_{01}, D_{01}, D_{10}, D_{00} have A-bases F_{11}(H′, H′), F_{01}(H′), F_{10}(H′), F_{00}(1) where for h′, k′ ∈ H′ := H \ 1 = {h ∈ H | |h| ⩾ 1}

F_{11}(h′, k′) := F_{h′,k′} := (h′)f(k′)^{∗} = h′ k′^{∗} − ∑_{x} xh′(xk′)^{∗},
F_{01}(k′) := F_{1,k′} := f(k′)^{∗} = k′^{∗} − ∑_{x} x(x^{∗}k′^{∗}),
F_{10}(h′) := F_{h′,1} := h′ f = h′ − ∑_{x} (h′x)x^{∗},
F_{00}(1) := F_{1,1} = 1 − ∑_{x} xx^{∗} = 1 − e = f.

3. The Scalar Multiple Theorem

The proper analogue of the Scalar Multiple Theorem [1, Thm.3.4 p.184], [6, Th.20.3 p.267] for the finite case is the following.

Theorem 3.1 (Scalar Multiple). If 2 ⩽ |X| < ∞, then every element of the deep matrix algebra DM_{X}(A) which doesn’t lie in the obstacle ideal Z_{X}(A) has a “scalar multiple”: if u ∈ DM_{X}(A) \ Z_{X}(A) then there exist a nonzero coordinate 0 ≠ α ∈ A and heads h, k ∈ H with

k^{∗}uh = α1, k^{∗}h = δ1 (δ = 1, 0).

Proof. ^{3} Let S denote the set D \ Z of deep matrices outside the obstacle. For u ∈ S we define the set [u] of proper multiples of u to be the set of all k^{∗}uh ∈ S. Note that v ∈ [u] ⟹ [v] ⊆ [u] since k_{2}^{∗}(k_{1}^{∗}uh_{1})h_{2} = (k_{1}k_{2})^{∗}u(h_{1}h_{2}). Our goal is to show that for a fixed element u ∈ S there exists a scalar in [u]. The natural choice is a v = ∑_{i=0}^{r} α_{i}h_{i}k_{i}^{∗} (α_{i} ≠ 0) in [u] with the fewest number r +1 of terms (so all h_{i}k_{i}^{∗} are distinct) and shallowest h_{0}k_{0}^{∗} (minimal m_{0} := |k_{0}| + |h_{0}|). We can shift v forwards or backwards by

(Shiftability) v ∈ [u] ⟹ ∃ x, y ∈ X such that vx, y^{∗}v ∈ [u].

Indeed, working modulo the ideal Z = DfD we have 0 ≢ v = ve + vf ≡ ve = ∑_{x∈X}(vx)x^{∗}, so vx ≢ 0 for at least one x. Analogously, 0 ≢ v = ev + fv ≡ ev = ∑_{y∈X} y(y^{∗}v), so y^{∗}v ≢ 0 for at least one y, and both multiples vx, y^{∗}v remain in S.

Since v was already shortest possible in [u], vx and y^{∗}v in S cannot have fewer terms. Thus α_{i}h_{i}k_{i}^{∗}x, α_{i}y^{∗}h_{i}k_{i}^{∗} ≠ 0. Then by D_{X} we know each k_{i} ≠ 1 must begin with x, each h_{i} ≠ 1 must begin with y. Moreover, the shallowest term must have h_{0} = k_{0} = 1: if k_{0} = xk′_{0} then h_{0}k_{0}^{∗}x = h_{0}k′_{0}^{∗} for m′_{0} = |h_{0}| + |k′_{0}| < |h_{0}| + |k_{0}| = m_{0}, contradicting minimality, and analogously h_{0} = yh′_{0} would lead to a shallower y^{∗}h_{0}k_{0}^{∗} = h′_{0}k_{0}^{∗}. Thus α_{0}1 is one of the r + 1 terms of v and all the other terms have

(∗) h_{i}k_{i}^{∗} ∈ yD + Dx^{∗} if i ≠ 0

[h_{i} = k_{i} = 1 would contradict distinctness of the h_{i}k_{i}^{∗}]. By Shiftability there is also z ∈ X with z^{∗}(vx) ∈ [u], so z^{∗}(α_{0}1)x ≠ 0 (by minimality) and therefore z = x. Hence x^{∗}(vx) = (x^{∗}v)x ∉ Z ⇒ x^{∗}v ∉ Z ⇒ x^{∗}v ∈ [u]. Thus we can take y = x from the start, so (∗) becomes

(∗∗) h_{i}k_{i}^{∗} ∈ xD + Dx^{∗} if i ≠ 0.

By our assumption |X| ⩾ 2 we can find w ≠ x in X, so w^{∗}(xD + Dx^{∗})w = 0 implies w^{∗}vw = w^{∗}(α_{0}1)w = α_{0}1 ∈ [u] (by the Non-Z Scalar Rule) as claimed. Thus the minimal r + 1 is 1 and there never were any other terms α_{i}h_{i}k_{i}^{∗} to begin with.

3Compare the more elaborate proof in [1, 2, Lemma 1.8, p.176], shifting u until it becomes k_{0}^{∗}uh_{0} = α1 + ∑_{h_{i}≠1} α_{h_{i}}h_{i} ∈ [u] for α ≠ 0 and then isolating the scalar by a pull μ_{p_{m}}^{∗} (as in Remark 1.2) [m > |k_{0}|, |h_{0}|, |h_{i}| for all i] killing all individual h and k^{∗} for nonempty heads h, k ≠ 1 of length < m and send the empty head 1 to 1. Cuntz created p_{m} using an infinite sequence x_{1}x_{2} … from X which was aperiodic; the simplest such sequence is x_{1}^{1}x_{2}x_{1}^{2}x_{2}x_{1}^{3}x_{2} … x_{1}^{n}x_{2} …, and p_{m} = x_{1}^{m}x_{2} for suitably large m has the desired properties μ_{p_{m}}^{∗}(hk^{∗}) = 0 for all |h|, |k| ⩽ m except μ_{p_{m}}^{∗}(x_{1}^{r} x_{1}^{r}^{∗}) = μ_{p_{m}}^{∗}(1) = 1 for 0 ⩽ r < m.

Finally, if k^{∗}h ≠ 0, 1 then the Head Product Rules show that k^{∗}h = h′ or k^{∗}h = h′^{∗} with |h′| ≠ 0. If h′ = xh′′ starts with x ∈ X choose y ≠ x (using |X| ⩾ 2 again) and replace h, k by the heads hy, ky. Then (ky)^{∗}u(hy) = y^{∗}(k^{∗}uh)y = α_{0}y^{∗}y = α_{0}1 again, but now (ky)^{∗}(hy) = y^{∗}k^{∗}hy becomes y^{∗}xh′′y = 0 or y^{∗}h′′^{∗}x^{∗}y = 0 as claimed. □

Another aspect of the obstacle (cf. also [4]) is that it is precisely the kernel of the Frankenstein action of DM_{X}(A) on the space V [B] of bodies. We have already noted that the obstacle is contained in the annihilator of the module, and now the reverse inclusion is an immediate consequence of the Scalar Multiple Theorem.

Theorem 3.2. When the index set X is finite, 2 ⩽ |X| < ∞, the kernel of the Frankenstein representation of DM_{X}(A) on the space V [B] of bodies is precisely the obstacle Z_{X}(A), and the reduced algebra \overline{DM_{X}(A)} = DM_{X}(A)/Z_{X}(A) ≅ F_{X}(A) is isomorphic to the Frankenstein algebra via \overline{F}(\overline{E_{h,k}}) = F_{h,k}.

Proof. If u is not in Z_{X}(A) we know by the Scalar Multiple Theorem that there is a scalar 0 ≠ α1 = k^{∗}uh, and since V [B] is a free A-module the nonzero scalar does not act trivially on V [B], so u cannot either, and u is not in the kernel. Thus the kernel cannot contain any more than Z_{X}(A), and the Frankenstein Epimorphism DM_{X}(A) ⟶^{F} F_{X}(A) has kernel precisely Z_{X}(A), so it induces an isomorphism \overline{F} of DM_{X}(A)/Z_{X}(A) with F_{X}(A). □

A corresponding result about scalar multiples of f holds even when |X| = 1.

Theorem 3.3 (Scalar f-Multiple). Every nonzero element u of the deep matrix algebra DM_{X}(A) for 1 ⩽ |X| < ∞ has a “scalar f-multiple”: there exist a nonzero coordinate 0 ≠ α ∈ A and heads h, k ∈ H with fk^{∗}uhf = αf.

Proof. The case |X| ⩾ 2 follows easily from the ordinary Scalar Multiple Theorem. If u ∉ Z then there is k^{∗}uh = α1 ≠ 0 for heads h, k, hence fk^{∗}uhf = f(α1)f = αf. So suppose u = ∑_{h,k} α_{h,k}F_{h,k} ∈ Z ≅ Mat_{∞}^{fin}(A). If α_{h_{0},k_{0}} = α ≠ 0, then by the F -Matrix Unit Rules αf = αF_{∅,∅} = F_{∅,h_{0}}uF_{k_{0},∅} = (fh_{0}^{∗})u(k_{0}f).

We give a general proof for all |X| ⩾ 1 along the lines of [6, Thm. 20.3 p.267]: some multiple v of u contains a nonzero term α 1, hence fvf = αf. Indeed, if u = ∑_{h,k} α_{h,k}hk^{∗} ≠ 0 for distinct hk^{∗}, let α_{h_{0},k_{0}} be a nonzero coefficient which is minimal in the sense that α_{h,k} = 0 when h ⊢ h_{0} is a proper initial segment, and also when h = h_{0} but k ⊢ k_{0} is a proper initial segment. Then the product fh_{0}^{∗} u k_{0}f has a unique term α_{h_{0},k_{0}}fh_{0}^{∗}(h_{0}k_{0}^{∗})k_{0}f = α_{h_{0},k_{0}}f1f = α_{h_{0},k_{0}}f since any other nonzero term α_{h,k}fh_{0}^{∗}(hk^{∗}) vanishes unless h_{0}, h are related by the Head Product Rule; we can’t have h ⊢ h_{0} by minimality, so we must have h_{0} ⊩ h = h_{0}h′, in which case the product fh_{0}^{∗}(h_{0}h′)k^{∗} = f h′k^{∗} vanishes unless h′ = 1, h = h_{0} (since f kills h′k′^{∗} if |h′| ⩾ 1 by (F1)). But when h = h_{0} we have fh_{0}^{∗}(hk^{∗})k_{0}f = fh_{0}^{∗}(h_{0}k^{∗})k_{0}f = f k^{∗}k_{0}f = 0 unless k, k_{0} are related; we can’t have α_{h_{0},k} ≠ 0 if k ⊢ k_{0} by minimality again, and k ≠ k_{0} by distinctness, so we must have k_{0} ⊢ k = k_{0}k′ for k′ ≠ 1, in which case f k^{∗}k_{0} f = f k′^{∗} f = 0 by (F3). □

Observe that when X is finite (so that e, f exist), on W we have ezb′ = 0, fzb′ = zb′ because in D′ we have f = zz^{∗} + f′ for f′ the new infinite idempotent f′.

4. The ideal lattice

For any ideal I ◁ A we have an ideal DM_{X}(I) := I DM_{X}(A) ◁ DM_{X}(A) with quotient DM_{X}(A)/DM_{X}(I) ≅ DM_{X}(A/I). For infinite X these are the only ideals of DM_{X}(A) [6, Thm 20.4 p.268] as with ordinary matrix algebras Mat_{n}^{fin}(A) and Mat_{∞}^{fin}(A), but for finite index sets X this is not quite the case, due to the obstacle. Clearly an ideal in DM_{X}(A) contains DM_{X}(I) iff it contains I1, and contains Z_{X}(I′) iff it contains I′f.

Theorem 4.1 (Ideal Lattice). The ideals of DM_{X}(A) are precisely all I = I_{I,I′} := DM_{X}(I) + Z_{X}(I′) for nested pairs I ⊆ I′ of ideals in A. Here I uniquely determines I, I′: I1 = I ∩ A1, Z_{X}(I′) = I ∩ Z_{X}(A) (equivalently, I′f = I ∩ Af).

In particular, if A is simple the only ideals are DM_{X}(A) = I_{A,A}, Z_{X}(A) = I_{0,A}, and 0 = I_{0,0}, so DM_{X}(A)/Z_{X}(A) and Z_{X}(A) are simple algebras.

Proof. Given I ◁ D, I ∩ A1 ◁ A1 must be I1 and by the Obstacle Theorem the ideal I ∩ Z ◁ Z must have the form Z_{X}(I′). Clearly I ⊆ I′ are ideals of A, and I ⊇ I1D + I′fZ = DM_{X}(I) + Z_{X}(I′) = I_{I,I′}. Note that I ∩ Z = Z_{X}(I′) implies I ∩ Af = I′f, so the two expressions for I′ are equivalent. We claim we have equality I = I_{I,I′}. Passing to \overline{I} ◁ \overline{D} = DM_{X}(A)/DM_{X}(I) ≅ DM_{X}(\overline{A}) for \overline{A} := A/I, if \overline{I} ⊈ \overline{Z} then by the Scalar Multiple Theorem we have a nonzero ᾱ1 ∈ \overline{I}; but then α1 ∈ I + DM_{X}(I) = I, forcing α ∈ I by the definition of I, hence ᾱ1 = ¯0, a contradiction. Thus we must have \overline{I} ⊆ \overline{Z} and I ⊆ DM_{X}(I) + Z_{X}(A). By the Dedekind Modular Law I = I ∩ (DM_{X}(I) + Z_{X}(A)) = DM_{X}(I) + I ∩ Z_{X}(A) = DM_{X}(I) + Z_{X}(I′) by definition of I′, as claimed. □

The algebra Mat_{n}^{fin}(A) arises as the endomorphisms of a free right A-module V_{A} ≅ A^{n}. Similarly, the Frankenstein algebra F = F_{X}(A) arises as endomorphisms of the free right Frankenstein module V [B] = BA, which becomes an (F, A)-bimodule. We also have a free left Frankenstein A-module V_{L}[B] = AB := ⨁_{b∈B} Ab and a representation of DM_{X}(A) as Φ-linear (but not A-linear unless A is commutative) endomorphisms of AB via (αE_{h,k})(βb) = αβE_{h,k}(b); again the annihilator of this module is precisely Z in the finite case, and V_{L}[B] provides a faithful representation for F. Instead of working with the left Frankenstein module V_{L}[B] as (F, Φ)-bimodule, we will work with the ordinary Frankenstein module V [B] as (F, A)-bimodule.

The sub-bimodules of the Frankenstein (F, A)-bimodule V [B] are described in [6, Th.20.10, p.272]. The argument given there works for arbitrary X and for (F, Φ)- bimodules as well. Recall that the tail-class τ of a body b is the equivalence class of b where b′ ∼ b if they have the same tail but different heads (b = kc, b′ = hc, equivalently b′ = hk^{∗}b). Thus the hk^{∗} act transitively on the members of a tail class, and for a fixed tail-class τ the tail submodule V_{τ} = ⊕_{b∈τ}bA is a cyclic F-module with any body b ∈ τ as generator. The Frankenstein module splits as a direct sum V [B] = ⨁_{τ} V_{τ} of these F-invariant tail submodules. For any ideal I ◁ A we obtain invariant submodules V_{τ}(I) := I V_{τ}(B) = V_{τ}(B) I = ⊕_{b∈τ}bI.

Theorem 4.2 (Frankenstein Submodules). The F_{X}(A)-invariant right A-submodules (the (F, A)-bimodules) of V [B] are precisely all

W = ⨁_{τ} V_{τ}(I_{τ}) for ideals I_{τ} := {α ∈ A | V_{τ}α ⊆ W }.

The irreducible sub-bimodules are the V_{τ}(I) for minimal ideals I ◁ A. If A is simple, V [B] is completely reducible with the irreducible invariant sub-bimodules precisely the V_{τ} = V_{τ}(A).

The F_{X}(A)-invariant right Φ-submodules (the (F, Φ)-bimodules) of V [B] are precisely all

W = ⨁_{τ} V_{τ}(L_{τ}) for left ideals L_{τ} := {α ∈ A | V_{τ}α ⊆ W }.

The irreducible submodules are the V_{τ}(L) for minimal left ideals L ◁_{ℓ} A. If A is completely reducible as a left A-module (a direct sum of simple left ideals) then V [B] is completely reducible as a left F-module. In particular, if A = Δ is a division algebra, V [B] is completely reducible with the irreducible F-invariant Φ-submodules precisely the V_{τ} = V_{τ}(Δ).

Proof. First notice that

I_{τ} = {α ∈ A | bAα ⊆ W for all b ∈ τ} L_{τ} = {α ∈ A | bAα ⊆ W for all b ∈ τ}
= {α ∈ A | b′α ∈ W for some b′ ∈ τ}  = {α ∈ A | b′α ∈ W for some b′ ∈ τ}

= {α ∈ A | bAαA ⊆ W for all b ∈ τ}

In both cases (bimodule or left module) the first equality is the definition, the second holds since bAα = Ahk^{∗}(b′α) [from b ∼ b′] and W is left-invariant under Ahk^{∗} ⊆ F, the third holds in the bimodule case since W is a right A-module. From the first equation it is clear that I_{τ} is a left ideal with W ⊇ ∑_{τ} V_{τ}(I_{τ}) = ∑_{b∈τ} bI_{τ} = ∑_{τ} V_{τ}I_{τ}, and from the third equation in the bimodule case it is clear that I_{τ} is an ideal, with ∑_{b∈τ} bI_{τ} = ∑_{τ} I_{τ}V_{τ} = ∑_{τ} V_{τ}I_{τ}.

Conversely, every w = ∑_{i} b_{i}α_{i} ∈ W for distinct bodies has each bα_{i} ∈ W: by Body Separation there is an e_{h} fixing b_{i} and killing the other b_{j}, hence b_{i}α_{i} = e_{h}(w) ∈ W. Then by the second criterion α_{i} ∈ I_{τ_{i}} for τ_{i} the tail-class of b_{i}, so b_{i}α_{i} ∈ V_{τ_{i}}(I_{τ_{i}}) for all i, and W ⊆ ∑_{τ} V_{τ}(I_{τ}). □

Similarly, the characterization of F-endomorphisms of the Frankenstein module [6, Th. 20.11, p.273] carries over verbatim.

Theorem 4.3 (Frankenstein Endomorphisms). The endomorphisms of V [B] as a right A-module which commute with the Frankenstein action (the (F_{X}(A), A)-bimodule endomorphisms) reduce to central scalar multiplications on the tail submodules:

\begin{matrix}End_{F_{X}(A)}(V [B]) = ⨁_{τ} Center(A)1_{V_{τ}} , \\ Hom_{F_{X}(A)}(V_{τ}, V_{τ}) = Center(A)1_{V_{τ}} , Hom_{F_{X}(A)}(V_{τ}, V_{σ}) = 0 (τ ≠ σ). \end{matrix}

Thus the V_{τ} are non-isomorphic (F_{X}(A), A)-bimodules.

Proof. First, each endomorphism φ acts diagonally: bA = ∩_{n=1}^{∞}e_{h_{n}(b)}V ⟹ φ(b) ∈ ∩_{i=1}^{∞}e_{h_{n}(b)}φ(V) ⊆ ∩_{n=1}^{∞}e_{h_{n}(b)}V = bA so that φ(b) = bα_{b} for each body. Since bodies with the same tail are conjugate under F, the scalars α_{b} are constant on tail-classes: if b′ ∼ b then b′ = hk^{∗}b and b′α_{b′} = φ(b′) = φ(hk^{∗}b) = hk^{∗}φ(b) = hk^{∗}bα_{b} = b′α_{b} implies α_{b′} = α_{b}. The diagonal scalars must lie in the center: φ(β1(b)) = φ(bβ) = φ(b)β [by right A-linearity] = bα_{b}β must equal β1(φ(b)) = β1(bα_{b}) = bβα_{b}, so α_{b} must commute with all β ∈ A.

Since any bimodule-homomorphism in Hom(V_{τ}, V_{σ}) extends to an endomorphism φ of V [B] by setting φ = 0 on all V_{ρ} for ρ ≠ τ; the the above diagonal result says φ = 0 if σ ≠ τ, and when σ = τ it is α1_{V_{τ}} for some central α. □

5. The Case |X| = 1

We are primarily concerned with the case |X| ⩾ 2, but we indicate here how the “exceptional case” [5] X = {x} of cardinality 1 becomes more complicated: the Frankenstein representation degenerates and the Kennedy representation becomes the standard Kaplansky shift representation, but the Scalar Multiple Theorem 3.1 fails and DM_{x}(A) has a complicated ideal structure.

Theorem 5.1 (Cardinality 1). If X = {x} has cardinality 1, then there is only one body b = (x, x, x, …), V [B] is 1-dimensional, F_{X}(A) = A1_{V [B]} is also 1-dimensional, and the Frankenstein ideal lattice is that of A.

On the other hand, V [H] = ⨁_{j⩾0} v_{j}A is infinite-dimensional with Kennedy representation the shift action, x(v_{j}) = v_{j+1} the forward shift operator, x^{∗}(v_{j}) = v_{j−1} (j ⩾ 1), x^{∗}(v_{0}) = 0 the backward shift. Thus DM_{x}(A) is faithfully imbedded in the rowand-column-finite matrices as the standard Kaplansky shift algebra via x → ∑_{j=0}^{∞} E_{j+1,j} with 1’s on the subdiagonal,and x^{∗} → ∑_{j=0}^{∞} E_{j,j+1} with 1’s on the superdiagonal. If A is simple this action is irreducible.

The reduced deep matrix algebra \overline{DM_{x}(A)} = DM_{x}(A)/Z_{x}(A) is isomorphic to the algebra A[t, t^{−1}] of Laurent polynomials over A, so the lattice of ideals of \overline{DM_{x}(A)} is isomorphic to the lattice of t-cancelable ideals of the polynomial ring A[t] (those I_{0} ◁ A[t] such that p(t) ∈ A[t], tp(t) ∈ I_{0} ⟹ p(t) ∈ I_{0}).

The Scalar Multiple Theorem fails in DM_{x}(A): there are no heads h, k satisfying k^{∗}(x + x^{∗})h = α1 ≠ 0.

Proof. This situation is polycephalic, with one body but lots of heads: there is only a single body b = (x, x, x, …) ∈ B(X), V [B] = bA is 1-dimensional, and the Frankenstein action is ( ∑ α_{m,n}x^{m}x^{∗}^{n})(b) = ( ∑_{m,n} α_{m,n})b, so the kernel of the Frankenstein Epimorphism is the set of deep matrices having coefficient sum zero, with quotient the Frankenstein algebra DM_{x}(A)/Ker(F) ≅ F_{X}(A) = A1_{V [B]}. Thus the Frankenstein algebra ideals are precisely those of A.

There are many heads: H(X) = {x^{n} | n ⩾ 0}, V [H] = ⨁_{n⩾0} x^{n}A =: ⨁_{n⩾0} v_{n}A is an infinite-dimensional free A-module with basis of v_{n} = x^{n}, and under the Kennedy representation DM_{x}(A) is isomorphic to the familiar Kaplansky shift algebra [3, Example 6(2) p.35] with x the forward shift x(v_{i}) = v_{i+1} and x^{∗} the backward shift x^{∗}(v_{i}) = v_{i−1}, x^{∗}(v_{0}) = 0. This algebra is spanned over A by the k^{th} subdiagonals x^{k} = ∑_{i=0}^{∞} E_{i+k,i} and superdiagonals x^{∗}^{k} = ∑_{i=0}^{∞} E_{i,i+k} (the 0^{th} sub- and superdiagonals being the identity matrix Id), plus Mat_{∞}^{fin}(Φ) spanned by the E_{i,j}. Indeed, computing the action on the v_{i}’s shows shows

\begin{matrix}x^{n}x^{∗}^{n} = ∑_{i=n}^{∞} E_{i,i}, x^{n+k}x^{∗}^{n} = ∑_{i=n}^{∞} E_{i+k,i}, x^{m}x^{∗}^{m+k} = ∑_{i=m}^{∞} E_{i,i+k}, \\ f = 1 − xx^{∗} = E_{0,0}, x^{∗}^{k}E_{0,0} = E_{0,0}x^{k} = 0 (k ⩾ 1), E_{i,j} = x^{i}E_{0,0}x^{∗}^{j}. \end{matrix}

These products show that Z_{x}(A) = AHH^{∗}fHH^{∗} = AHfH^{∗} = ∑_{i,j⩾0} Ax^{i}fx^{∗}^{j} = ∑_{i,j⩾0} AE_{i,j} ≅ Mat_{∞}^{fin}(A), so the reduced deep matrix algebra DM_{x}(A)/Z_{x}(A) = ( ∑_{n=1}^{∞} Ax^{∗}^{n} + A + ∑_{m=1}^{∞} Ax^{m}) + Z_{x}(A) with x^{∗}x = 1, xx^{∗} = 1 − f ∈ 1 + Z_{x}(A) is isomorphic to the algebra A[t, t^{−1}] = ∑_{k=−∞}^{∞} At^{k} of Laurent polynomials over A mapping the cosets of x, x^{∗} to t, t^{−1} with t^{−1}t = tt^{−1} = 1. Since DM_{x}(A)/Ker(F) ≅ A, the Frankenstein kernel is much bigger than the obstacle Z_{x}(A).

It is easily checked that the ideals I ◁ A[t, t^{−1}] are precisely all I = ∑_{k=0}^{∞} t^{−k}I_{0} for t-cancelable ideals I_{0} = I ∩ A[t] ◁ A[t]. But it seems difficult to describe these for arbitrary associative coordinate rings A.

For heads h = x^{n}, k = x^{m}, k^{∗}(x + x^{∗})h = x^{∗}^{m}x^{n+1} + x^{∗}^{m+1}x^{n} has a term α1 only if m = n + 1 or m + 1 = n, yielding 1 + x^{∗}^{2} or x^{2} + 1, so we cannot reduce to a single term and we can never obtain k^{∗}uh = α1 ≠ 0. The trouble with a single x is lack of orthogonality: no product of x’s and x^{∗}’s is ever zero. □

References

[1] J. Cuntz: Simple C^{∗}-algebras generated by isometries. Comm. in Math. Phys. 57: 173-185 (1970).

[2] J. Cuntz, W. Krieger: A class of C^{∗}-algebras and topological Markov chains. Inv. Math. 56: 251-268 (1980).

[3] N. Jacobson: Structure of Rings, Amer. Math.Soc. Pub. vol. 37, Amer. Math. Soc., Providence, 1956.

[4] C. Kennedy: Deep matrix algebras of finite type, Algebras and Representation Theory 9: 525-537 (2006).

[5] L. Kennedy: Exceptional type deep matrix algebras, in progress.

[6] K. McCrimmon: Chapter 20: Deep matrices and their Frankenstein actions, in Nonassociative Algebras and its Applications by L. Sabinen, L. Sbitneva, I. Shestakov (eds.), Lecture Notes in Pure and Applied Mathematics, vol. 246, CRC Press, Boca Raton (2006).

Department of Mathematics, University of Virginia, Charlottesville, VA 22903 U.S.A.