Tensor Algebras overC*-Correspondences: Representations, Dilations, andC*-Envelopes

Operator Algebras and C*-correspondences: A Survey

“…The tensor algebra T + X of a C * -correspondence A X A is the norm-closed subalgebra of T X generated by all elements of the form π ∞ (a), t n ∞ (ξ), a ∈ A,ξ ∈ X n , n ∈ N. The tensor algebras for C * -correspondences were pioneered by Muhly and Solel in [40]. They form a broad class of non-selfadjoint operator algebras which includes as special cases Peters' semicrossed products [45], Popescu's non-commutative disc algebras [47], the tensor algebras of graphs (introduced in [40] and further studied in [29]) and the tensor algebras for multivariable dynamics [13], to mention but a few. For more examples, see [31].…”

“…Then, O G is the C * -algebra generated by a 'maximal' direct sum of such families. It turns out that there O G is the Cuntz-Pimsner algebra of a certain C * -correspondence [40]. The associated Cuntz-Pimsner -Toeplitz algebra is the universal algebra for the first four relations in ( †) and is denoted as T G .…”

Topological freeness for Hilbert bimodules

“…Thus potentially, by passing to the core C * -algebra, see [1, Thm. 3.1], it may be applied to all the C * -algebras equipped with a semi-saturated circle action and thereby to all relative CuntzPimsner algebras [13]. As a corollary of our main result we provide an ideal lattice description (in the case the dual system is free) and a simplicity criterion for the algebras considered.…”

A note on the gauge invariant uniqueness theorem for C*-correspondences

“…There is a strong connection between T X and O X attained by Muhly and Solel [15] and Fowler, Muhly and Raeburn [4] under certain assumptions, and settled in full generality by Katsoulis and Kribs [8]. The tensor algebra T + X in the sense of Muhly and Solel [15] is the non-involutive closed subalgebra of T X generated by A and X.…”

“…The tensor algebra T + X in the sense of Muhly and Solel [15] is the non-involutive closed subalgebra of T X generated by A and X. Then O X is the minimal C*-cover of T + X , i.e., the C*-envelope of T + X in the sense of Arveson [1].…”