A Morita Characterisation for Algebras and Spaces of Operators on Hilbert Spaces
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Integral Equations and Operator Theory
A Morita Characterisation for Algebras and Spaces of Operators on Hilbert Spaces G. K. Eleftherakis and E. Papapetros Abstract. We introduce the notion of Δ and σ Δ− pairs for operator algebras and characterise Δ− pairs through their categories of left operator modules over these algebras. Furthermore, we introduce the notion of Δ-Morita equivalent operator spaces and prove a similar theorem about their algebraic extensions. We prove that σΔ-Morita equivalent operator spaces are stably isomorphic and vice versa. Finally, we study unital operator spaces, emphasising their left (resp. right) multiplier algebras, and prove theorems that refer to Δ-Morita equivalence of their algebraic extensions. Mathematics Subject Classification. Primary: 47L30, Secondary: 47L25, 46M15. Keywords. Operator algebras, C ∗ -Algebras, TRO, Stable isomorphism, Morita equivalence.
1. Introduction In what follows, we denote by B(H1 , H2 ) the space of all linear and bounded operators from the Hilbert space H1 to the Hilbert space H2 . If X is a subset of B(H1 , H2 ) and Y is a subset of B(H2 , H3 ), then we denote by [YX ] the norm-closure of the linear span of the set {y x ∈ B(H1 , H3 ) , y ∈ Y , x ∈ X } . Similarly, if Z is a subset of B(H3 , H4 ), we define the space [ZYX ]. A linear subspace M ⊆ B(H, K) is called a ternary ring of operators (TRO) if M M M ⊆ M. It then follows that M is an A − B equivalence bimodule in the sense of Rieffel for the C -algebras A = [M M ] and B = [M M ]. 0123456789().: V,-vol
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We call a norm closed ternary ring of operators M, σ-TRO if there exist sequences {mi ∈ M , i ∈ N} and {nj ∈ M , j ∈ N} such that lim t
and
t
mi mi m = m, lim
i=1
t
t
m nj nj = m, ∀ m ∈ M
j=1
t t mi mi ≤ 1 , nj nj ≤ 1 , ∀ t ∈ N. i=1
j=1
Equivalently, a TRO M is a σ-TRO if and only if the C -algebras [M M ], [M M ] have a σ-unit. At the beginning of the 1970s, M. A. Rieffel introduced the idea of Morita equivalence of C -algebras. In particular, he gave the following definitions: (i) Two C -algebras, A and B, are said to be Morita equivalent if they have equivalent categories of -representations via -functors. (ii) The same algebras are said to be strongly Morita equivalent if there exists an A − B module of equivalence or if there exists a TRO M such that the C algebras [M M ] and A (resp. [M M ] and B) are isomorphic. We write A ∼R B. If A ∼R B, then A and B have equivalent categories of representations. The converse does not hold. For further details, see [15–17]. Brown, Green and Rieffel proved the following fundamental theorem for C algebras ([7,8]). Theorem 1.1. If A , B are C -algebras with σ-units, then A ∼R B if and only if they are stably isomorphic, which means that the algebras A ⊗ K , B ⊗ K are -isomorphic. Here, K is the algebra of compact operators acting on 2 (N), and ⊗ is the minimal tensor product. The next step in this t
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