P. Ara, R. Hazrat, H. Li, and A. Sims, Graded Steinberg algebras and their representations, Algebr. Number Theory, vol.12, issue.1, pp.131-172, 2018.
DOI : 10.2140/ant.2018.12.131

URL : http://arxiv.org/pdf/1704.01214

P. Ara, M. Frías, and E. Pardo, Nonstable K-theory for graph algebras, Algebr. Represent. Theory, vol.10, issue.2, pp.157-178, 2007.
DOI : 10.1007/s10468-006-9044-z

URL : http://arxiv.org/pdf/math/0412243

K. Stephen-austin and A. Mitra, Groupoid models of C * -algebras and Gelfand duality, 2018.

M. Beattie, A generalization of the smash product of a graded ring, J. Pure Appl. Algebra, vol.52, issue.3, pp.219-226, 1988.

M. Viviane, L. Beuter, and . Cordeiro, The dynamics of partial inverse semigroup actions, p.31, 2018.
URL : https://hal.archives-ouvertes.fr/ensl-01956972

M. Viviane, D. Beuter, and . Gonçalves, The interplay between Steinberg algebras and skew rings, J. Algebra, vol.497, pp.337-362, 2018.

M. Viviane, D. Beuter, J. Gonçalves, D. Öinert, and . Royer, Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics, Forum Math, vol.31, issue.3, pp.543-562, 2019.

A. Buss and R. Exel, Inverse semigroup expansions and their actions on C * -algebras, Illinois J. Math, vol.56, issue.4, pp.1185-1212, 2012.

M. Toke, J. Carlsen, and . Rout, Diagonal-preserving graded isomorphisms of Steinberg algebras, Commun. Contemp. Math, vol.20, issue.6, pp.1750064-1750065, 2018.

L. Clark and C. , Uniqueness theorems for Steinberg algebras, vol.18, pp.907-916, 2015.

L. Clark, R. Exel, E. Pardo, A. Sims, and C. Starling, Simplicity of algebras associated to non-Hausdorff groupoids, 2018.

L. Clark, C. Farthing, A. Sims, and M. L. Tomforde, A groupoid generalisation of Leavitt path algebras, Semigroup Forum, vol.89, issue.3, pp.501-517, 2014.

M. Cohen and S. Montgomery, Group-graded rings, smash products, and group actions, Trans. Amer. Math. Soc, vol.282, issue.1, pp.237-258, 1984.
DOI : 10.1090/s0002-9947-1984-0728711-4

URL : https://www.ams.org/tran/1984-282-01/S0002-9947-1984-0728711-4/S0002-9947-1984-0728711-4.pdf

G. Luiz and . Cordeiro, Étale inverse semigroupoids -the fundamentals, 2019.

H. Martín, R. Escardó, and . Heckmann, Topologies on spaces of continuous functions, Topol. Proc, vol.26, pp.545-564, 2001.

R. Exel, Circle actions on C * -algebras, partial automorphisms, and a generalized PimsnerVoiculescu exact sequence, Bull. Braz. Math. Soc. (N.S.), vol.122, issue.2, pp.191-313, 1994.

R. Exel and F. Vieira, Actions of inverse semigroups arising from partial actions of groups, J. Math. Anal. Appl, vol.363, issue.1, pp.86-96, 2010.

J. Fell, An extension of Mackey's method to Banach * -algebraic bundles, Memoirs of the, Representations of * -algebras, locally compact groups, and Banach * -algebraic bundles, vol.2, 1969.

T. Giordano, I. F. Putnam, and C. F. Skau, Topological orbit equivalence and C * -crossed products, J. für die reine und Angew. Math. (Crelle's Journal), vol.469, issue.1, pp.285-320, 1995.

R. Godement, Sur la théorie des représentations unitaires, Ann. of Math, vol.53, issue.1, pp.68-124, 1951.

I. Kaplansky, The structure of certain operator algebras, Trans. Amer. Math. Soc, vol.70, issue.2, pp.219-255, 1951.

A. Kelarev, Semisimple rings graded by inverse semigroups, J. Algebra, vol.205, issue.2, pp.451-459, 1998.

A. Anthony-kumjian, Fell bundles over groupoids, Proc. Amer. Math. Soc, vol.126, issue.4, pp.1115-1125, 1998.

M. Verus, L. , and D. Lenz, Pseudogroups and their étale groupoids, Adv. Math, vol.244, pp.117-170, 2013.

X. Shao, F. Liu, and . Van-oystaeyen, Group graded rings, smash products and additive categories, Perspectives in ring theory, vol.233, pp.299-310, 1987.

V. Liu, Free inverse semigroupoids and their inverse subsemigroupoids, p.85, 2016.

P. Lundström, Crossed product algebras defined by separable extensions, J. Algebra, vol.283, issue.2, pp.723-737, 2005.

, Víctor Eduardo Marín Colorado and Héctor Edonis Pinedo Tapia, Partial groupoid actions on categories: globalization and the smash product

K. Paul-mcclanahan, K-theory for partial crossed products by discrete groups, J. Funct. Anal, vol.130, issue.1, pp.77-117, 1995.

. Walter-douglas-munn, Rings graded by inverse semigroups, Semigroups (Proceedings Int. Conf. Semigroups held Univ. Minho, pp.136-145, 1999.

C. N?st?asescu and F. Van-oystaeyen, Methods of graded rings, Lecture Notes in Mathematics, vol.1836, 2004.

P. Nystedt and J. Öinert, Simple semigroup graded rings, J. Algebr. Its Appl, vol.14, issue.7, pp.1550102-1550103, 2015.

A. Leonard-tuke-paterson, Groupoids, inverse semigroups, and their operator algebras, vol.170, 1999.

D. Quinn, Group-graded rings and duality, vol.292, pp.155-167, 1985.

S. W. Rigby, Tensor products of Steinberg algebras, 2018.

. Walter-rudin, Nándor Sieben, C * -crossed products by partial actions and actions of inverse semigroups, Real and complex analysis, vol.40, pp.32-46, 1987.

B. Steinberg, Diagonal-preserving isomorphisms of étale groupoid algebras, J. Aust. Math. Soc, vol.223, issue.2, pp.412-439, 2010.

B. R. Tilson, Categories as algebra: an essential ingredient in the theory of monoids, J. Pure Appl. Algebra, vol.48, issue.1-2, pp.83-198, 1987.

M. Louis-tomforde, Uniqueness theorems and ideal structure for Leavitt path algebras, J. Pure Appl. Algebra, vol.318, issue.1, pp.471-484, 2007.

. John-von-neumann, On rings of operators. Reduction theory, Ann. of Math, vol.50, issue.2, p.69364, 1949.

, Lyon Cedex 07, France E-mail address: luizgc6@gmail.com, luis-gustavo